Cutting *only* existing geometry with a circle
I will assume that you know how to use the Knife Project tool but if not, there are plenty of tutorials on BSE that can show you.  

Here's a simple technique that might help you achieve what you're after:

Get everything ready to use the Knife Project tool but just before you do, select all of the vertices of the shape you want cut, and add them to a Vertex Group.  Once you do that, cut your shape out with the Knife Project tool.  You can then Deselect the vertices from your original Vertex Group and use the Dissolve Vertices feature to get rid of the unwanted vertices that you added with Knife Project: